I am having trouble understanding the Sum rule in genetics with \"at least\" pro

Question

I am having trouble understanding the Sum rule in genetics with "at least" problems.

Here is a question with an answer already. Can you explain how I would get the answer if she asked for at least one with a family of 4 children instead of 3?

Sum rule A couple decide to have three children. What is the probability of having a family with at least one girl? Child one Child two Child three probability of that family 1/8 1/8 girl-½ boy - 1/2 girl-½ girl- girl-½ 1/8 Etc ! Sum rule can be used as follows: what is the only type of family that does not have at least one girl? What is the probability of that family? Thus 1-1/8 7/8

Explanation / Answer

Genetically there is always 50% chance of having a girl or boy child. Hence, as mentioned in your question when 3 children were expected there is 1/2 chance of getting a girl or boy child every time. Similarly, when 4 children would be expected there will be again 50% chance. Previously when 3 children were expected probability for that family was 1/2*1/2*1*2= 1/8. And probability of getting at least one girl child would be 1-1/8= 7/8. We come to this conclusion according to the sum rule which states that the probability of any of the various mutually exclusive events that is going to occur is equivalent to the sum of events. So, here since we are talking about having kids, there is chance of getting either a girl or a boy child. This is a mutually exclusive event. Both cannot be achieved at the same time. So probability of getting a boy child is 1/2 and probability of getting a girl child is also 1/2.

If 4 children are expected the probability would be 1/2*1/2*1/2*1/2=1/16

And since one girl is expected, the probability would be 1-1/16=15/16

We consider it as 1-1/16 because the probability of occurrence of any event is considered as 1 and probability of non-occurence of any event is 0. So, every time we will deduct the probability from 1. That would be the probability of having at least one girl child how much you increase the number of children.

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